### 数学代写|编码理论代写Coding theory代考|Basics of Coding Theory

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## 数学代写|编码理论代写Coding theory代考|Finite Fields

Finite fields play an essential role in coding theory. The theory and construction of finite fields can be found, for example, in [1254] and [1408, Chapter 2]. Finite fields, as related specifically to codes, are described in [1008, 1323, 1602]. In this section we give a brief introduction.

Definition 1.2.1 A field $\mathbb{F}$ is a nonempty set with two binary operations, denoted $+$ and $\cdot$, satisfying the following properties.
(a) For all $\alpha, \beta, \gamma \in \mathbb{F}, \alpha+\beta \in \mathbb{F}, \alpha \cdot \beta \in \mathbb{F}, \alpha+\beta=\beta+\alpha, \alpha \cdot \beta=\beta \cdot \alpha, \alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma$, $\alpha \cdot(\beta \cdot \gamma)=(\alpha \cdot \beta) \cdot \gamma$, and $\alpha \cdot(\beta+\gamma)=\alpha \cdot \beta+\alpha \cdot \gamma$.
(b) $\mathbb{F}$ possesses an additive identity or zero, denoted 0 , and a multiplicative identity or unity, denoted 1 , such that $\alpha+0=\alpha$ and $\alpha \cdot 1=\alpha$ for all $\alpha \in \mathbb{F}_{q}$.
(c) For all $\alpha \in \mathbb{F}$ and all $\beta \in \mathbb{F}$ with $\beta \neq 0$, there exists $\alpha^{\prime} \in \mathbb{F}$, called the additive inverse of $\alpha$, and $\beta^{} \in \mathbb{F}$, called the multiplicative inverse of $\beta$, such that $\alpha+\alpha^{\prime}=0$ and $\beta \cdot \beta^{}=1$.

The additive inverse of $\alpha$ will be denoted $-\alpha$, and the multiplicative inverse of $\beta$ will be denoted $\beta^{-1}$. Usually the multiplication operation will be suppressed; that is, $\alpha \cdot \beta$ will be denoted $\alpha \beta$. If $n$ is a positive integer and $\alpha \in \mathbb{F}, n \alpha=\alpha+\alpha+\cdots+\alpha\left(n\right.$ times ), $\alpha^{n}=\alpha \alpha \cdots \alpha$ ( $n$ times), and $\alpha^{-n}=\alpha^{-1} \alpha^{-1} \cdots \alpha^{-1}$ ( $n$ times when $\alpha \neq 0$ ). Also $\alpha^{0}=1$ if $\alpha \neq 0$. The usual rules of exponentiation hold. If $\mathbb{F}$ is a finite set with $q$ elements, $\mathbb{F}$ is called a finite field of order $q$ and denoted $\mathbb{F}_{q}$.

Example 1.2.2 Fields include the rational numbers $\mathbb{Q}$, the real numbers $\mathbb{R}$, and the complex numbers $\mathbb{C}$. Finite fields include $\mathbb{Z}_{p}$, the set of integers modulo $p$, where $p$ is a prime.

## 数学代写|编码理论代写Coding theory代考|Generator and Parity Check Matrices

When choosing between linear and nonlinear codes, the added algebraic structure of linear codes often makes them easier to describe and use. Generally, a linear code is defined by giving either a generator or a parity check matrix.

Definition 1.4.1 Let $\mathcal{C}$ be an $[n, k]{q}$ linear code. A generator matrix $G$ for $\mathcal{C}$ is any $G \in \mathbb{F}{q}^{k \times n}$ whose row span is $\mathcal{C}$. Because any $k$-dimensional subspace of $\mathbb{F}{q}^{n}$ is the kernel of some linear transformation from $\mathbb{F}{q}^{n}$ onto $\mathbb{F}{q}^{n-k}$, there exists $H \in \mathbb{F}{q}^{(n-k) \times n}$, with independent rows, such that $\mathcal{C}=\left{\mathbf{c} \in \mathbb{F}_{q}^{n} \mid H^{\mathrm{T}}=\mathbf{0}^{\mathrm{T}}\right}$. Such a matrix, of which there are generally many, is called a parity check matrix of $\mathcal{C}$.

Example 1.4.2 Continuing with Example 1.3.2, there are several generator matrices for $\mathcal{C}{1}$ including $$G{1}=\left[\begin{array}{llll} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 \ 0 & 0 & 1 & 1 \end{array}\right], G_{1}^{\prime}=\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \ 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \end{array}\right] \text {, and } G_{1}^{\prime \prime}=\left[\begin{array}{cccc} 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 1 & 1 \end{array}\right] \text {. }$$
In this case there is only one parity check matrix $H_{1}=\left[\begin{array}{lll}1 & 1 & 1\end{array}\right.$
Remark 1.4.3 Any matrix obtained by elementary row operations from a generator matrix for a code remains a generator matrix of that code.

Remark 1.4.4 By Definition 1.4.1, the rows of $G$ form a basis of $\mathcal{C}$, and the rows of $H$ are independent. At times, the requirement may be relaxed so that the rows of $G$ are only required to span $\mathcal{C}$. Similarly, the requirement that the rows of $H$ be independent may be dropped as long as $\mathcal{C}=\left{\mathbf{c} \in \mathbb{F}_{q}^{n} \mid H \mathbf{c}^{\top}=\mathbf{0}^{\mathrm{T}}\right}$ remains true.

## 数学代写|编码理论代写Coding theory代考|Distance and Weight

The error-correcting capability of a code is keyed directly to the concepts of Hamming distance and Hamming weight. ${ }^{3}$

Definition 1.6.1 The (Hamming) distance between two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{F}{q}^{n}$, denoted $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})$, is the number of coordinates in which $\mathbf{x}$ and $\mathbf{y}$ differ. The (Hamming) weight of $\mathbf{x} \in \mathbb{F}{q}^{n}$, denoted $w t{\mathrm{H}}(\mathbf{x})$, is the number of coordinates in which $\mathbf{x}$ is nonzero.
Theorem 1.6.2 ([1008, Chapter 1.4]) The following hold.
(a) (nonnegativity) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y}) \geq 0$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{F}{q}^{n}$.
(b) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})=0$ if and only if $\mathbf{x}=\mathbf{y}$. (c) (symmetry) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})=\mathrm{d}{\mathrm{H}}(\mathbf{y}, \mathbf{x})$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{F}{q}^{n}$.
(d) (triangle inequality) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{z}) \leq \mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})+\mathrm{d}{\mathrm{H}}(\mathbf{y}, \mathbf{z})$ for all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{F}{q}^{n}$.
(e) $\mathrm{d}{\mathrm{H}}(\mathbf{x}, \mathbf{y})=\mathrm{wt}{\mathrm{H}}(\mathbf{x}-\mathbf{y})$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{F}{q}^{n}$. (f) If $\mathbf{x}, \mathbf{y} \in \mathbb{F}{2}^{n}$, then
where $\mathbf{x} \star \mathbf{y}$ is the vector in $\mathbb{F}{2}^{n}$ which has $1 s$ precisely in those coordinates where both $\mathbf{x}$ and $\mathbf{y}$ have $1 s$. (g) If $\mathbf{x}, \mathbf{y} \in \mathbb{F}{2}^{n}$, then $\mathrm{wt}{\mathrm{H}}(\mathbf{x} \star \mathbf{y}) \equiv \mathbf{x} \cdot \mathbf{y}(\bmod 2)$. In particular, $\mathrm{wt}{\mathrm{H}}(\mathbf{x}) \equiv \mathbf{x} \cdot \mathbf{x}(\bmod 2)$.

(h) If $\mathbf{x} \in \mathbb{F}{3}^{n}$, then $\mathrm{wt}{\mathrm{H}}(\mathbf{x}) \equiv \mathbf{x} \cdot \mathbf{x}(\bmod 3)$.
Remark 1.6.3 A distance function on a vector space that satisfies parts (a) through (d) of Theorem 1.6.2 is called a metric; thus $\mathrm{d}_{\mathrm{H}}$ is termed the Hamming metric. Other metrics useful in coding theory are examined in Chapter $22 .$

Definition 1.6.4 Let $\mathcal{C}$ be an $(n, M){q}$ code with $M>1$. The minimum (Hamming) distance of $\mathcal{C}$ is the smallest distance between distinct codewords. If the minimum distance $d$ of $\mathcal{C}$ is known, $\mathcal{C}$ is denoted an $(n, M, d){q}$ code (or an $[n, k, d]{q}$ code if $\mathcal{C}$ is linear of dimension $k$ ). The (Hamming) distance distribution or inner distribution of $\mathcal{C}$ is the list $B{0}(\mathcal{C}), B_{1}(\mathcal{C}), \ldots, B_{n}(\mathcal{C})$ where, for $0 \leq i \leq n$,
$$B_{i}(\mathcal{C})=\frac{1}{M} \sum_{\mathbf{c} \in \mathcal{C}}\left|\left{\mathbf{v} \in \mathcal{C} \mid \mathrm{d}_{\mathrm{H}}(\mathbf{v}, \mathbf{c})=i\right}\right|$$

## 数学代写|编码理论代写Coding theory代考|Finite Fields

(a) 对所有人一个,b,C∈F,一个+b∈F,一个⋅b∈F,一个+b=b+一个,一个⋅b=b⋅一个,一个+(b+C)=(一个+b)+C, 一个⋅(b⋅C)=(一个⋅b)⋅C， 和一个⋅(b+C)=一个⋅b+一个⋅C.
(二)F拥有一个加法单位或零，表示为 0 ，和一个乘法单位或单位，表示为 1 ，使得一个+0=一个和一个⋅1=一个对所有人一个∈Fq.
(c) 对所有人一个∈F和所有b∈F和b≠0， 那里存在一个′∈F，称为加法逆一个， 和b∈F，称为乘法逆b, 这样一个+一个′=0和b⋅b=1.

## 数学代写|编码理论代写Coding theory代考|Generator and Parity Check Matrices

1001 0101 0011\right], G_{1}^{\prime}=\left[

1111 1100 0110\right] \text { 和 } G_{1}^{\prime \prime}=\left[

1100 0110 0011\右] \文本{。}


## 数学代写|编码理论代写Coding theory代考|Distance and Weight

(a)（非消极性）dH(X,是)≥0对所有人X,是∈Fqn.
(二)dH(X,是)=0当且仅当X=是. (c) (对称)dH(X,是)=dH(是,X)对所有人X,是∈Fqn.
(d) (三角不等式)dH(X,和)≤dH(X,是)+dH(是,和)对所有人X,是,和∈Fqn.
（和）dH(X,是)=在吨H(X−是)对所有人X,是∈Fqn. (f) 如果X,是∈F2n，然后

(h) 如果 $\mathbf{x} \in \mathbb{F} {3}^{n},吨H和n\mathrm{wt} {\mathrm{H}}(\mathbf{x}) \equiv \mathbf{x} \cdot \mathbf{x}(\bmod 3).R和米一个rķ1.6.3一个d一世s吨一个nC和F在nC吨一世○n○n一个在和C吨○rsp一个C和吨H一个吨s一个吨一世sF一世和sp一个r吨s(一个)吨Hr○在GH(d)○F吨H和○r和米1.6.2一世sC一个ll和d一个米和吨r一世C;吨H在s\ mathrm {d} _ {\ mathrm {H}一世s吨和r米和d吨H和H一个米米一世nG米和吨r一世C.○吨H和r米和吨r一世Cs在s和F在l一世nC○d一世nG吨H和○r是一个r和和X一个米一世n和d一世nCH一个p吨和r22 .$

B_{i}(\mathcal{C})=\frac{1}{M}\sum_{\mathbf{c}\in \mathcal{C}}\left|\left{\mathbf{v}\in\数学{C} \mid \mathrm{d}_{\mathrm{H}}(\mathbf{v}, \mathbf{c})=i\right}\right|B_{i}(\mathcal{C})=\frac{1}{M}\sum_{\mathbf{c}\in \mathcal{C}}\left|\left{\mathbf{v}\in\数学{C} \mid \mathrm{d}_{\mathrm{H}}(\mathbf{v}, \mathbf{c})=i\right}\right|

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## MATLAB代写

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